Editing Three utilities problem (section)

===Changing the rules===
{{multiple image|total_width=480
|image1=3 utilities problem moebius.svg|caption1=Solution on a Möbius strip
|image2=3_utilities_problem_torus.svg|caption2=Solution on a torus
|image3=4_utilities_problem_torus.svg|caption3=A torus allows up to 4 utilities and 4 houses
}}
<math>K_{3,3}</math> is a [[toroidal graph]], which means that it can be embedded without crossings on a [[torus]], a surface of genus one.{{r|harary}} These embeddings solve versions of the puzzle in which the houses and companies are drawn on a [[coffee mug]] or other such surface instead of a flat plane.{{r|parker}} There is even enough additional freedom on the torus to solve a version of the puzzle with four houses and four utilities.{{r|obeirne|early}} Similarly, if the three utilities puzzle is presented on a sheet of a transparent material, it may be solved after twisting and gluing the sheet to form a [[Möbius strip]].{{r|larsen}}

Another way of changing the rules of the puzzle that would make it solvable, suggested by [[Henry Dudeney]], is to allow utility lines to pass through other houses or utilities than the ones they connect.{{r|dud17}}